The core of most real-time optimization (RTO) schemes consists of a parameter estimation problem and an economic optimization problem. Typically, the purpose of the RTO system is to determine the optimal steady-state of the process being optimized using process measurements and a mathematical model of the process, which may or may not include an approximation of an intermediate control layer. Thus, RTO operates on a timescale that is considerably slower than most lower-level controllers—-an iteration is usually completed once every hour to once every few days. Because of this relative abundance of computational time, both the parameter estimation problem and the economic optimization problem are often formulated as non-convex nonlinear programs (NLPs), which more accurately represent the process over large operating regions, but are more difficult to solve, than the linear or quadratic programs common in some lower-level control schemes (Marlin and Hrymak, 1996).

However, even though the models used are fairly accurate, RTO systems are still subject to significant uncertainties. Indeed, the processes being modeled are typically very complex and some approximations must be made in order to obtain a tractable model. In addition, while a well-designed parameter estimation problem does allow the model to adapt to changing conditions, this problem relies on measurements, which are typically subject to noise, finite precision, etc. Thus, the resulting estimates will not be exact and should be considered to be stochastic in nature.

Efforts have been made to account for such uncertainties in RTO systems. While the community is aware of techniques for tackling the semi-infinite nonlinear programming problem that arises when one asks for the NLP constraints to hold for all possible values of the parameters in some deterministic set, like those of Darlington et al., 1999, and Romagnoli et al., 1996, these techniques have not taken hold, presumably because of their computational complexity and non-convexity (Contreras-Dordelly and Marlin 2000, Marlin 2006). In particular, these techniques involve the iterative solution of several NLPs: one large program where the constraints are required to hold for many discrete values of the parameters, and a series of smaller programs, one for each uncertain inequality constraint. While there is typically a significant amount of computational time available to RTO systems, such an approach may still be out of reach because a) the underlying NLP may be quite large to begin with (on the order of 50 control variables and thousands of nonlinear equations (Marlin and Hrymak, 1996)) and b) the smaller nonlinear programs must be solved to global optimality, at least in the last iteration, which is a condition that cannot be guaranteed for general non-convex programs.

Thus, in the RTO literature, the problem of optimization under uncertainty is almost always simplified via linearization of the constraints. It is also common to calculate “backoffs” for the inequality constraints—-positive quantities added to the nominal values so as to ensure that the constraints will be satisfied even in the presence of disturbances. Often multiple backoff terms are calculated for each constraint, some deterministic and some stochastic. The methods for calculating the stochastic backoffs may be placed into two categories: explicit probabilistic constraints (Loeblein and Perkins, 1998, Zhang et al., 2002) and Monte Carlo sampling of the uncertain parameters. The latter category includes at least two distinct approaches. In the first, Monte Carlo sampling is used to calculate the probability of constraint violation (considering all of the uncertain inequality constraints at once) along with the derivatives of this probability with respect to the optimization variables (Zhang et al., 2002). In the second, a series of parameter values are generated at which all of the (linearized) constraints are required to hold (Contreras-Dordelly and Marlin 2000).

Proposed Approach

Here it is proposed to calculate RTO backoffs using the first-order robust formulation by Zhang, 2006. In many ways this method may be seen as a compromise between the full semi-infinite approach and the RTO approaches outlined above. With the former it shares the characteristic of finding a solution that satisfies the constraints for all values of the parameters in some uncertainty set; but like the latter methods, the constraints are first linearized. Zhang further assumes that the uncertainty set is defined by some ball about the nominal value of the parameters; it is this assumption that makes the method particularly attractive. In particular, it allows one to determine an explicit formulation for the robust constraints and to derive a bound on the maximum amount of violation of the nonlinear version of the inequality constraints, assuming that the uncertainty set is accurate. Specifically, this bound is shown to be proportional to a Lipschitz constant times the square of a scalar measure of the uncertainty set size.

The first-order robust formulation appears to be particularly well suited for RTO systems for at least four reasons. First of all, the uncertainty sets assumed by Zhang fit naturally with the RTO parameter estimation problem. In particular, it is straightforward to calculate box or ellipsoidal confidence regions for the parameters using data from the parameter estimation problem, and such regions are valid uncertainty sets. Secondly, the first-order robust formulation results in a single, relatively moderately sized NLP which should be computationally advantageous compared to the full semi-infinite approach or the Monte Carlo methods. In particular, if there are n dependent variables and p parameters, then there will be just np additional variables and equality constraints. Thirdly, while the probability of a constraint violation is not specified, a bound on the maximum amount of constraint violation is given, and mere feasibility, not global optimality, is required in order for this bound to hold. Related to this, the bound is proportional to the size of the uncertainty set squared-—thus if the RTO parameter estimation scheme is able to keep the uncertainty levels moderate, we would expect small violations, if any, in the inequality constraints, at least compared to more uncertain problems like those encountered in process design. Finally, even though complete feasibility cannot be guaranteed, the repetitive nature of RTO means that, in the case that constraint violations are observed, the Lipschitz constant entering the bound just described may be estimated and used to inflate the uncertainty sets in order to all but eliminate constraint violations, if that is desirable (Hale and Zhang, 2006).

Case Study

This approach to RTO backoff calculations is demonstrated using the gasoline blending example in Zhang et al., 2002. While this system is somewhat simple because there is no specified controller operating under the RTO layer, this example is still sufficient for comparing various backoff calculations. In particular, the formulation of the backoff is independent of whether or not a model of a control layer is included in the economic optimization—-the inclusion of such a model merely results in a reshuffling of the variable categorization (the original independent variables become dependent variables and the new independent variables are the controller set-points), and reduces the amount of backoff necessary assuming that the controller is able to compensate for a significant amount of the parameter variations, as was seen in Contreras-Dordelly and Marlin 2000.

Two scenarios will be considered. In the first, the uncertain parameters and their covariance are estimated using maximum likelihood estimation, and the resulting uncertainty sets are used in the nonlinear programming formulation that results from applying Zhang's methodology to the economic optimization problem subject to the full nonlinear model. The second scenario is similar, but the original economic optimization problem is formulated by linearizing the nonlinear model about the control variables but not the parameters. The result is a second-order cone program. These results are compared to those obtained by Zhang et al., who applied chance-constraint programming to the fully linearized model (linearized about the control variables and the parameters) via both explicit probabilistic constraints and Monte Carlo simulation.

Contreras-Dordelly, J. L. and T. E. Marlin. Control design for increased profit. Comput. Chem. Eng., 24: 267-272, 2000.

Darlington, J., C. C. Pantelides, B. Rustem and B. A. Tanyi. An algorithm for constrained nonlinear optimization under uncertainty. Automatica, 35:217-228, 1999.

Hale, E. T. and Y. Zhang. Case studies for a first-order robust nonlinear programming formulation. In preparation.

Loeblein, C. and J. D. Perkins. Economic analysis of different structures of on-line process optimization systems. Comput. Chem. Eng., 22(9):1257-1269, 1998.

Marlin, T. E. Personal communication. April 21, 2006.

Marlin, T. E. and A. N. Hrymak. Real-time operations optimization of continuous processes. In Kantor, J. C., C. E. Garca and B. Carnahan, editors, 5th International Conference on Chemical Process Control, volume 93 of AIChE Symposium Series. American Institute of Chemical Engineers, 1997.

Romagnoli, J. A., J. Bandoni and A. Parisa. Effect of disturbances in optimizing control : steady-state open-loop back-off problem. AIChE J., 42(4): 983-994, 1996.

Zhang, Y. A General Robust-Optimization Formulation for Nonlinear Programming. To appear in J. Optimization Theory and Applications, 2006.

Zhang, Y., D. Monder and J. F. Forbes. Real-time optimization under parametric uncertainty: a probability constrained approach. J. Proc. Cont., 12:373-389, 2002.